610 Mr. J. H. Jeans on the 



Jn particular, the number for which K (defined by equation 

 13) lies between K a and K 2 will be 



K=K 2 



%...e-^d^dh (16) 



where the integration extends through those parts of the sub- 

 sidiary space for which K lies between Ki and K 2 . 



§ 22. Now in the subsidiary space K will be found to have a 



minimum value. In equation (13), let us treat ( — j as a con- 



tinuous variable, and replace it by 6. We then have 



The values of 6 are limited by the condition 

 ^ a s N 



or what is the same thing, by 



-20,= 1 (18) 



n 8 



For small variations in 0, the condition that K shall be a 

 minimum is seen by variation of equations (17) and (18) to be 



-2{l + log0, + \}80. = O, .... (19) 



71 s 



where X is an indeterminate multiplier. We must therefore 

 have 



l+log0. + A = O ...... (20) 



tor all values of s, and hence S = constant. From equation 

 (18) we see that this constant value must be S = 1. Thus 

 when we regard as/a as a continuous variable the minimum 

 occurs when a x — a 2 = . . . = a s = . . . = a Q . When a s is very 

 great, and only capable of integral values, the minimum 

 still occurs for values such that, except for infinitesimally 

 small quantities, 



& i &2 ®s -| / n -i \ 



The minimum value of K is found to be K = 0, except for 

 an infinitesimally small quantity : it does not, however, 

 follow that NK = 0. The minimum value of K subject to 



